A full Eulerian finite difference approach for solving fluid-structure coupling problems

Journal of Computational Physics

Volumn 230, Issue 3, 2011, Pages 596-627

  Sugiyama, Kazuyasu ^a   Ii, Satoshi ^a   Takeuchi, Shintaro ^a   Takagi, Shu ^a,b   Matsumoto, Yoichiro ^a  

^a UNIVERSITY OF TOKYO   (Japan)

^b RIKEN   (Japan)

Author keywords

Eulerian formulation; Finite difference method; Fluid structure interaction; Hyperelastic material; Volume of fluid

Indexed keywords

FINITE DIFFERENCE METHOD; FLUID STRUCTURE INTERACTION; NONLINEAR EQUATIONS; NUMERICAL METHODS;

BASIC EQUATIONS; COUPLING PROBLEM; EULERIAN; EULERIAN FORMULATIONS; FINITE DIFFERENCE APPROACH; FINITE-DIFFERENCE METHODS; FLUID-STRUCTURE COUPLING; FLUID-STRUCTURE INTERACTION; HYPERELASTIC MATERIALS; VOLUME OF FLUIDS;

CONSTITUTIVE EQUATIONS;

EID: 78649325897     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.09.032     Document Type: Article

References (97)

1. Amsden A.A., Harlow F.H. A simplified MAC technique for incompressible fluid flow calculations. J. Comput. Phys. 1970, 6:322-325.
2. Bazilevs Y., Calo V.M., Zhang Y., Hughes T.J.R. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput. Mech. 2006, 38:310-322.
3. Belytschko T. Fluid-structure interaction. Comput. Struct. 1980, 12:459-469.
4. Bonet J., Wood R.D. Nonlinear Continuum Mechanics for Finite Element Analysis 2008, Cambridge University Press, Cambridge, (Chapter 4). second ed.
5. Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral Methods in Fluid Dynamics 1988, Springer-Verlag, New York, (Chapter 4).
6. Chang Y.C., Hou T.Y., Merriman B., Osher S. A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. 1996, 124:449-464.
7. Chen J.-S., Pan C. A pressure projection method for nearly incompressible rubber hyperelasticity, Part I: Theory. Trans. ASME J. Appl. Mech. 1996, 63:862-868.
8. Cottet G.-H., Maitre E., Milcent T. Eulerian formulation and level set models for incompressible fluid-structure interaction. Math. Model. Numer. Anal. 2008, 42:471-492.
9. Dunne T. An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaptation. Int. J. Numer. Methods Fluids 2006, 51:1017-1039.
10. Eggleton C.D., Popel A.S. Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 1998, 10:1834-1845.
11. Figueroa C.A., Vignon-Clemental I.E., Jansen K.E., Hughes T.J.R., Taylor C.A. A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput. Methods Appl. Mech. Eng. 2006, 195:5685-5706.
12. Gao T., Hu H.H. Deformation of elastic particles in viscous shear flow. J. Comput. Phys. 2009, 228:2132-2151.
13. Gil A.J. Structural analysis of prestressed Saint Venant-Kirchhoff hyperelastic membranes subjected to moderate strains. Comput. Struct. 2006, 84:387-411.
14. Glowinski R., Pan T.-W., Hesla T.I., Joseph D.D. A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 1999, 25:755-794.
15. Glowinski R., Pan T.-W., Hesla T.I., Joseph D.D., Périaux J. A ficticious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 2001, 169:363-426.
16. Gong X., Sugiyama K., Takagi S., Matsumoto S. The deformation behavior of multiple red blood cells in a capillary vessel. J. Biomech. Eng. 2009, 131:074504.
17. Gueyffier D., Li J., Nadim A., Scardovelli R., Zaleski S. Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 1999, 152:423-456.
18. Gültop T. On the propagation of acceleration waves in incompressible hyperelastic solids. J. Sound Vib. 2003, 264:377-389.
19. Gurtin M.E., Guidugli P.P. The thermodynamics of constrained materials. Arch. Ration. Mech. Anal. 1973, 51:192-208.
20. Hamamoto M., Hisada T., Suzumori K. Feasibility study of an electrostatic actuator by finite element interaction analysis. Int. J. Appl. Electromagnet. Mech. 1999, 10:249-257.
21. Harlow F.H., Welch J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 1965, 8:2182-2189.
22. Hirt C.W., Amsden A.A., Cook J.L. An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 1974, 14:227-253.
23. Hirt C.W., Nichols B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39:201-225.
24. T. Hisada, Foundations of Tensor Analysis for Nonlinear Finite Element Method, 1992 (Chapters 2 and 5, in Japanese, Maruzen).
25. Hu H.H. Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 1996, 22:335-352.
26. Huang H., Sugiyama K., Takagi S. An immersed boundary method for restricted diffusion with permeable interfaces. J. Comput. Phys. 2009, 228:5317-5322.
27. Huang W.-X., Sung J. An immersed boundary method for fluid-flexible structure interaction. Comput. Methods Appl. Mech. Eng. 2009, 198:2650-2661.
28. Huerta A., Liu W.K. Viscous flow with large free surface motion. Comput. Methods Appl. Mech. Eng. 1988, 69:277-324.
29. Huerta A., Liu W.K. Viscous flow structure interaction. Trans. ASME J. Pressure Vessel Technol. 1988, 110:15-21.
30. Hughes T.J.R., Liu W.K., Zimmermann T.K. Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 1981, 29:329-349.
31. Hughes T.J.R., Stewart J.R. A space-time formulation for multiscale phenomena. J. Comput. Appl. Math. 1996, 74:217-229.
32. S. Ii, K. Sugiyama, S. Takeuchi, S. Takagi, Y. Matsumoto, Development of accurate numerical model for the fluid-structure interaction problem based on Eulerian framework, in: Proceedings of the 10th US National Cong. on Comput. Mech., Ohio, USA, 2009, No. 158294.
33. S. Ii, K. Sugiyama, S. Takeuchi, S. Takagi, Y. Matsumoto, An implicit full Eulerian method for the fluid-structure interaction problem, Int. J. Numer. Methods Fluids, (2010), (in press). doi:10.1002/fld.2460.
34. Ikeno T., Kajishima T. Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations. J. Comput. Phys. 2007, 226:1485-1508.
35. Jacqmin D. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 1999, 155:96-127.
36. Jiang G.-S., Shu C.-W. Efficient implementation of weighted ENO scheme. J. Comput. Phys. 1996, 126:202-228.
37. Johnson A.A., Tezduyar T.E. 3D simulation of fluid-particle interactions with the number of particles reaching 100. Comput. Methods Appl. Mech. Eng. 1997, 145:301-321.
38. Kajishima T. Conservation properties of finite difference method for convection. Trans. Jpn. Soc. Mech. Eng. B 1994, 60-574:2058-2063. (in Japanese).
39. Kajishima T., Takiguchi S. Interaction between particle clusters and particle-induced turbulence. Int. J. Heat Fluid Flow 2002, 23:639-646.
40. Kajishima T., Takiguchi S., Hamasaki H., Miyake Y. Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding. JSME Int. J. Ser. B 2001, 44:526-535.
41. LeVeque R.J., Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 1994, 31:1019-1044.
42. Li Z., Ito K. The Immersed Interface Method 2006, SIAM, Philadelphia.
43. Li Z., Lai M.-C. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys. 2001, 171:822-842.
44. Liu W.K., Liu Y., Farrell D., Zhang L., Wang X.S., Fukui Y., Patankar N., Zhang Y., Bajaj C. Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Eng. 2006, 195:1722-1749.
45. Liu X.-D., Osher S., Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994, 115:200-212.
46. Liu C., Walkington N.J. An Eulerian description of fluids containing visco-elastic particles. Arch. Ration. Mech. Anal. 2001, 159:229-252.
47. Matsunaga N., Liu H., Himeno R. An imgage-based computational fluid dynamics method for haemodynamic simulation. JSME Int. J. Ser. C 2002, 45:989-996.
48. Mooney M. A theory of large elastic deformation. J. Appl. Phys. 1940, 11:582-592.
49. Mori Y., Peskin C.S. Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Eng. 2008, 197:2049-2067.
50. Nitikitpaiboon C., Bathe K.J. An arbitrary Lagrangian-Eulerian velocity potential formulation for fluid-structure interaction. Comput. Struct. 1993, 47:871-891.
51. S. Noda, K. Fukasaku, R. Himeno, Blood flow simulator using medical images without mesh generation, in: IFMBE Proceedings of World Congress on Medical Physics and Biomedical Engineering 2006, Seoul, Korea, 2006, pp. 36-40.
52. Noh W.F., Woodward P.R. SLIC (simple line interface calculation). Lecture Notes in Physics 1976, 59:330-340.
53. H. Okada, S.N. Atluri, Embedded localized strain zone constitutive model in finite strain and finite rotation, in: Proceedings of International Conference on Computational Engineering Science, vol. 2, 1995, pp. 2154-5159.
54. Okazawa S., Kashiyama K., Kaneko Y. Eulerian formulation using stabilized finite element method for large deformation solid dynamics. Int. J. Numer. Methods Eng. 2007, 72:1544-1559.
55. Osher S., Fedkiw R. Level set methods: an overview and some recent results. J. Comput. Phys. 2001, 169:463-502.
56. Osher S., Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces 2003, Springer-Verlag, New York.
57. Peskin C.S. Flow patterns around heart valves: a numerical method. J. Comput. Phys. 1972, 10:252-271.
58. Peskin C.S. The immersed boundary method. Acta Numer. 2002, 11:479-517.
59. Pozrikidis C. Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 2001, 440:269-291.
60. Pozrikidis C. Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 2005, 17:031503.
61. Rivlin R.S. Large elastic deformations of isotropic materials IV, further development of general theory. Phil. Trans. Royal Soc. A 1948, 241:379-397.
62. Saad Y., Schultz Y. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1986, 7:856-869.
63. Sawada T., Hisada T. Fluid-structure interaction analysis of the two-dimensional flag-in-wind problem by an interface-tracking ALE finite element method. Comput. Fluids 2007, 36:136-146.
64. Sethian J.A., Smereka P. Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 2003, 35:341-372.
65. Shi X., Phan-Thien N. Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions. J. Comput. Phys. 2005, 206:81-94.
66. Simo J.C., Taylor R.L., Pisler K.S. Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 1985, 51:177-208.
67. Skotheim J.M., Mahadevan L. Soft lubrication: the elastohydrodynamics of nonconforming and conforming contacts. Phys. Fluids 2005, 17:092101.
68. Stein K., Benney R., Tezduyar T., Potvin J. Fluid-structure interactions of a cross parachute: numerical simulation. Comput. Methods Appl. Mech. Eng. 2001, 191:673-687.
69. Sugiyama K., Ii S., Takeuchi S., Takagi S., Matsumoto Y. Full Eulerian simulations of biconcave neo-Hookean particles in a Poiseuille flow. Comput. Mech. 2010, 46:147-157.
70. Sussman M., Smereka P., Osher S. A level set approach for computating solutions to incompressible two-phase flow. J. Comput. Phys. 1994, 114:146-159.
71. Takagi S., Oguz H.N., Prosperetti A. PHYSALIS: a new method for particle simulation: part II: two-dimensional Navier-Stokes flow around cylinders. J. Comput. Phys. 2003, 187:371-390.
72. Takeuchi S., Yuki Y., Ueyama A., Kajishima T. A conservative momentum exchange algorithm for interaction problem between fluid and deformable particles. Int. J. Numer. Methods Fluids 2010, (in print). 10.1002/fld.2272.
73. Tan Z., Lim K.M., Khoo B.C. An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries. J. Comput. Phys. 2009, 228:6855-6881.
74. Taylor C.A., Hughes T.J.R., Zarins C.K. Finite element modeling of blood flow in arteries. Comput. Methods. Appl. Mech. Eng. 1998, 158:155-196.
75. Tezduyar T.E., Behr M., Liou J. A new strategy for finite element computations involving moving boundaries and interfaces - the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests. Comput. Methods Appl. Mech. Eng. 1992, 94:339-351.
76. Tezduyar T.E., Behr M., Mittal S. A new strategy for finite element computations involving moving boundaries and interfaces - the deforming-spatial-domain/space-time procedure: II. Computations of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput. Methods Appl. Mech. Eng. 1992, 94:353-371.
77. Torii R., Oshima M., Kobayashi T., Takagi K. Numerical simulation system for blood flow in the cerebral artery using CT imaging data. JSME Int. J. Ser. C 2001, 44:982-989.
78. Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T.E. Influence of wall elasticity in patient-specific hemodynamics simulations. Comput. Fluids 2007, 36:160-168.
79. Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T.E. Fluid-structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling. Comput. Mech. 2008, 43:151-159.
80. Trapp J.A. Reinforced materials with thermo-mechanical constraints. Int. J. Eng. Sci. 1971, 9:757-773.
81. Tryggvason G., Sussman M., Hussaini M.Y. Immersed boundary methods for fluid interfaces. Computational Methods for Multiphase Flow 2007, Cambridge University Press, Cambridge, (Chapter 3). A. Prosperetti, G. Tryggvason (Eds.).
82. Udaykumar H.S., Tran L., Belk D.M., Vanden K.J. An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces. J. Comput. Phys. 2003, 186:136-177.
83. Van Hoogstraten P.A.A., Slaats P.M.A., Baaijens F.P.T. A Eulerian approach to the finite element modelling of neo-Hookean rubber material. Appl. Sci. Res. 1991, 48:193-210.
84. Watanabe H., Sugiura S., Kafuku H., Hisada T. Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method. Biophys. J. 2004, 87:2074-2085.
85. Wheeler A.A., Boettinger W.J., McFadden G.B. A phase-field model for isothermal phase transitions in binary alloys. Phys. Rev. A 1992, 45:7424-7439.
86. Xiao F., Honma Y., Kono T. A simple algebraic interface capturing scheme using hyperbolic tangent function. Int. J. Numer. Methods Fluids 2005, 48:1023-1040.
87. Xiao F., Yabe T. Computation of complex flows containing rheological bodies. Comput. Fluid Dyn. J. 1999, 8:43-49.
88. Yokoi K., Xiao F., Liu H., Fukasaku K. Three-dimensional numerical simulation of flows with complex geometries in a regular Cartesian grid and its application to blood flow in cerebral artery with multiple aneurysms. J. Comput. Phys. 2005, 202:1-19.
89. Youngs D.L. Time-dependent multi-material flow with large fluid distortion. Numerical Methods for Fluid Dynamics 1982, 273-285. Academic Press, New York. K.W. Morton, M.J. Baines (Eds.).
90. Yu Z. A DLM/FD method for fluid/flexible-body interactions. J. Comput. Phys. 2005, 207:1-27.
91. Yuki Y., Takeuchi S., Kajishima T. Efficient immersed boundary method for strong interaction problem of arbitrary shape object with self-induced flow. J. Fluid Sci. Technol. 2007, 2:1-11.
92. Zhang L.T., Gay M. Immersed finite element method for fluid-structure interactions. J. Fluids Struct. 2007, 23:839-857.
93. Zhang L.T., Gay M. Imposing rigidity constraints on immersed objects in unsteady fluid flows. Comput. Mech. 2008, 42:357-370.
94. Zhang L., Gerstenbetger A., Wang X., Liu W.K. Immersed finite element method. Comput. Methods Appl. Mech. Eng. 2004, 193:2051-2067.
95. Zhang Q., Hisada T. Analysis of fluid-structure interaction problems with structural buckling and large domain changes by ALE finite element method. Comput. Methods Appl. Mech. Eng. 2001, 190:6341-6357.
96. Zhang Z., Prosperetti A. A second-order method for three-dimensional particle simulation. J. Comput. Phys. 2005, 210:292-324.
97. Zhao H., Freund J.B., Moser R.D. A fixed-mesh method for incompressible flow-structure systems with finite solid deformation. J. Comput. Phys. 2008, 227:3114-3140.